Improving achievable rates of concatenated GKP codes for Gaussian displacement channel using analog decoding

The coherent information of a Gaussian displacement noise channel is known to be an achievable rate with the use of Gottesman-Kitaev-Preskill (GKP) codes for certain values of displacement spread. The CSS Hamming bound for concatenation of square GKP with CSS codes falls short of the coherent information. However, the use of analog information of the lower-level GKP codes allows one to cross the hamming bound. We demonstrate this advantage by explicitly showing an achievable rate for any displacement noise by concatenating GKP codes with a finite-rate discrete variable code. We then proceed to compare these achievable rates with the coherent information of Gaussian displacement noise. This can potentially be used to find good multimode GKP lattices.